Aurino,
patterns exist in mathematics and nature on various levels. that is partly how we are able to relate the two and then manipulate them to either give us useful information or do useful things with them. it takes our imagination to percieve those patterns.
the other side of the coin is we can first imagine patterns and then contrive mathematics and nature to fit those imagined patterns.
i'll give you this much, those patterns that we percieve in mathematics and nature may or may not be the whole story, that is as we look deeper the patterns may dissolve and tranform into some other pattern or apparently patternless form.
i've only skimmed Dr. Dick's paper in totality and am currently reviewing in depth the first chapter. so i can't answer your question, "So what if you can artificially "fill the gaps" in a set of numbers? What does that have to do with anything?" with respect to Dr. Dick's paper.
from my own (hopefully not to fuzzy common sense) i'd say that if in general you can fill the gaps in an arbitray set of numbers and you can relate that set of numbers to reality then the values of those gaps may represent unknown information about reality, hence the potential for predicting or knowing something unknown about reality would be accessble to you.